Mike Kaechele

Category: math

  • Math Icebreaker

    Lots of teachers use icebreakers to start off the year. I use some with mixed results. Sometimes I feel like they are a waste of time. I like group problem-solving activities better than “find someone with the same birthday month as you” type.  Well I had an idea for a way to start off this year at my construction job this week.This idea is in the WCYDWT philosophy but is more hands on than video based. I also spell out the problem as a challenge.

    At my job we are forming walls on a funky house right now. It is two rectangles crossed in an X. The angle of the intersection is 60 degrees. It was not laid out by surveyors so we had to set stakes to find the exact dimensions and angles. It was quite a challenge. (I had to laugh when the homeowner, a computer programmer, pulled out his compass app on his iPhone 4 to check our pins. He also wanted the main part of the house parallel to a fence row 300 feet away with hills and trees in between.)

    So my “icebreaker” is to give groups of students six wooden stakes, a tape measure, and a hammer. I will also allow them to use calculators, textbooks, and perhaps their phones as resources. The problem is for them to lay out a perfect rectangle with an area of 48 square feet.

    I like this problem because it is cooperative, real world, outside, hands-on, and has multiple ways to solve it. I picked 48 square feet because students can use a 6-8-10 (3-4-5) triangle to find 90 degrees. They also can check the diagonals for congruency to find out if it is square. Students can also just use trial and error to try to make their rectangles better.

    I see this as a 9th grade level or higher problem. It allows for review of Pythagorean Theorem and properties of rectangles. Of course discussion of different solution strategies and how to “prove” a rectangle is perfect at the end are a critical part of making this activity successful.

  • How I laid out the square

    Here goes my attempt at explaining how we laid out a square parallel to our school around a circle with diameter of 21′. I must confess that I did the math and started by having some students help me in class. I ended up having two students stay after school with me for about an hour to finish the layout.
    First,I may have been unclear in the previous post but the circle already existed that the square needed to go around. So we measured 12′ 6″ off the building in two spots outside of the circle to establish the east line of the square parallel to the building. We put stakes in and ran a string line. Next we measured over 21′ from that line to establish the west side of the square also parallel to the building. That was the easy part. The tricky part was finding the corners.

    (The building is on the east side and west is up on this sketch)

    Our reference points were the furthermost northern and southern points of the circle. These points are the midpoints of the north and south sides of the square. But how did we establish  another point to create a line that it is perpendicular to the east line? If you are thinking Pythagorean Theorem that is part of it, but how do you find the corner? You guess, of course. Mathematicians, in the real world sometimes you have to estimate.

    We multiplied a 3-4-5 triangle by 3 to get the dimensions 9′-12′-15′. We measured 12′ off from the east string in line with the midpoint. Then we held a string from the east string to the west string so that it barely touched the stake and we “eyeballed” it square with those lines. We marked the intersection between this string and the east string with a sharpie on the east string. We measured over from this mark 9′ and put another mark on the east string. Next we measured from this mark to the stake (the hypotenuse) and it should have been 15′. Of course we guessed on our corner point so we were off one inch and it measured 15′ 1″.  We corrected this by shifting the corner and this point north 1″. Now the hypotenuse measured exactly 15′ and we put a stake in at the northeast corner.

    Now it gets easier. We pulled a string from the established northeast corner to the west line so that it barely touched the stake at 12′. This allowed us a very good estimate of the northwest corner. Next we measured over 21′ from both northern corners on our string lines to establish the two southern corners.

    Our last step was to measure the diagonals which should be congruent. We ended up about an inch off so we double checked our overall dimensions and found one stake was leaning in so it was not quite 21′. We straightened it and our diagonals were within 1/2″.

    The next day we measured over 2′ 6″ from all four sides of our square and pulled lines to find their intersections which gave us the four corners of our outer square. I really do not think that this problem was solvable by my 8th grade class but I wish I would have taken the time to walk them all through it.I think it was a great opportunity to show my students that math does apply to real problems in blue-collar jobs not just at universities or in a lab somewhere.

    PS If I was doing it over I would have just “picked” center point on the east line pulled off the building where it touched the circle. Then I could have measured 10′ 6″ both ways and established both of the east corners and used Pythagorean theorem to find perpendicular lines off from both corners. It would have been much easier 🙂